One of the reasons that boosters with the same vaccine work so well at regaining efficacy with new strains is not just because they prime the immune system into a ready state, but because while you're exposed to an antigen the dendritic cells are constantly presenting said antigens to new B-cells inside your lymphatic system. Those B-cells all have random antibodies and even where those antibodies are effective against the same antigen they're going to have slightly different structures which are all going to have different levels of affinity even to altered antigens.
Our adaptive immune system is not a perfect surgical strike. It's a carpet bomb which basically tries to place bets on every number on the roulette wheel. The more time an antigen is presented, the higher the chance you have of generating an antibody which overwhelms the virus. Nobody takes "perfect is the enemy of the good" more seriously than our immune systems. The immune system will go for mass amounts of "good" every time. One antibody that's good for one antigen may also be perfect for a slightly different one which is why your body keeps a lot of them around.
Look at a live vaccine like MMR. It's a live attenuated vaccine and it does actually live inside our body for the duration of our lives although we're not sure where. You only need one working shot to confer lifelong immunity because the attenuated virus keeps reproducing and getting presented in the lymphatic system to your new B-cells your entire life. They keep creating new antibodies throughout your entire life and the sheer overwhelming number of antibodies your body has on file against a measles/mumps/rubella antigen keeps you basically invincible from them. The only reason we do two shots is just in case something was wrong with the first shot and so we don't have to mass test for follow-up immunity.
mRNA on the other hand is short lived inside the body. Its half-life is about 10 hours before it basically shatters back into its constituent bases inside our cells' cytoplasm. Once that mRNA is gone that's all your body is going to produce of the spike protein. Once all the spike proteins have been produced, taken up, and presented, your B-cells aren't going to get any more exposure or training. A booster kicks that all up into high gear all over again and your antibody selection gets even broader. With a broader selection of antibodies available, the chance that you harbor antibodies with a high enough affinity to the next SARS-CoV-2 antigen in the wild, whatever the variant, goes up.
All infection stuff is about probability and dice rolls which messes with our idiot monkey brains that crave certainty. Keeping yourself safe from infection you can basically reduce the number of dice rolls, you can increase the number of dice you roll in defense, and you can increase the amount of numbers that can cause you to win instead of lose. Boosters do the latter two in spades.